Relation

 Relation:

                A relation R over a set X can be seen as a set of ordered pairs (x, y) of members of X.

    

               X= {1,2,3,4,5,6,7}

               the relation "is less than" on the set X is given by R= {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)}

                If we consider social relationships like fatherhood, brotherhood, sonhood, etc. 

                For example, Ram is the son of Dashrath, Sita was the daughter of Janak, etc.

Types of Relations:

• Empty Relation: 

                    A relation R on a set A is called Empty if the set A is empty set.

• Reflexive Relation:

                 A relation R on a set A is called reflexive if (a, a) € R holds for every element a € A i.e., if set A = {a, b} then R = {(a, a), (b, b)} is reflexive relation.

• Irreflexive relation:

                A relation R on a set A is called irreflexive if no (a, a) € R holds for every element a € A i.e., if set A = {a, b} then R = {(a, b), (b, a)} is irreflexive relation. 

• Symmetric Relation:

                A relation R on a set A is called symmetric if (b,a) € R holds when (a,b) € R.i.e. The relation R= {(4,5), (5,4), (6,5), (5,6)} on set A= {4,5,6} is symmetric.

• Transitive Relation:

                A relation R on a set A is called transitive if (a,b) € R and (b,c) € R then (a,c) € R for all a,b,c € A.i.e. Relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.

• Asymmetric relation:

                Asymmetric relation is opposite of symmetric relation. A relation R on a set A is called asymmetric if no (b,a) € R when (a,b) € R.

Set

 Set:

         A "set" is an undefined term,but we define it as a collection of well defined objects.

        e.g. A=collection of all the students in class 5 .

P= Collection of keys in a keychain.

Etc...

Types of sets:

Null set:

      A set having no elements is known as null set.

Singleton set:

      The set having only one element is known as singleton set.

Representation of set:

   There are two ways to represent a set 

    1-Tabular form

    2-Roster form

 1-Tabular Form:

      In this form of representation sets are basically represented by capital letters (A,B,C....Z), the elements are represented by small letters (a,b,c...z) and the elements are present within a curli braces "{}" and separated by commas","  .

      A={a,b,c}  ,  P={1,2,3}  etc.

2-Roster Form:

    There are some sets which are difficult to represent in tabular form as the number of elements are vast. like the set of real numbers,sets of rational etc.

    These sets are easily representable by the help of   Roster form .

  e.g.   R={x: x is a rela number}

           N={x: x is a natural number}

Subset:

  If there are 2 sets and all the elements of the 1st set are in the 2nd one the we say that the 1st set is the subset of the 2nd one .

            A={a,b,c,d}

            B={a,b,c,d,e,f}

  Then we say that ,   A ⊆ B

And we use the symbol "⊂" to represent the proper subset.

Mathematically,

    If A ⊆ B,

    Then  x ∈ A => x ∈ B

Power set:

    The set of all the subsets of a set is called as power set of that particular set.

 e.g.

A={a,b,c}

then power set of A 

P(A)={ { },{a},{b},{c},{a,b],{a,c},{b,c},{a,b,c} }

if the number of elements in a set is "n" then the number of elements on it`s power set is 2^n.

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 Number System:-

     Basically there are 2 type of number system 

              1-Real number System.

              2-Complex number System.

 Real number system:

        The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers. 

      There are different types of real numbers such as,

1 Natural numbers(N)={1,2,3...}

2 Whole numbers(W)={0,1,2,3...}

3 Integers(Z) ={...-3,-2,-1,0,+1,+2,+3...}

4 Rational numbers(Q)(p/q,q!= 0,p,q belongs to Z)

5 Irrational numbers(Q')(numbers which are not rational)

 

 Below is a diagram of real number system

                   

Complex Number System:

         Numbers which can be represented in the form of a+ib(i is a imaginary number,i=√-1)& a,b belongs to Z.

                   

Probability

Probability:

            Probability is the study of random or nondeterministic events.
  e.g.  if we toss a fair coin, then it results either a head or tail; of course we cannot be sure of the outcome in advance.
              N=the number of tosses
              n=the number of heads of obtained,
            P(n)=1/2

Statistical Experiment:

       A random or statistical experiment is one in which
i) all possible outcomes of the experiment are known in advance.
ii) A performance of an experiment results in an outcome which is not known in advance.
iii) the experiment can be repeated under identical conditions.

Sample space:

    The sample space of an experiment is the set of all possible outcomes of the experiment.

   e.g. if we toss a coin twice,the sample space will be 

        S={hh,ht,th,tt}

Event:

    A subset of a sample space is called an event. An event is said to occur if an element of the event occurs

   e.g. E={hh,ht}

Prob. of an event:

       If the sample space S is finite, the probability of an event A denoted by P(A) is defined as 

                              P(A)=size of A/size of S